<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">r</font> be   non  <a href="xxreal_0.html#V3" title="XXREAL_0:attr.3">negative</a>   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/>  for <font color="Olive" title="b2">o</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Olive" title="b1">n</font></span>)</span><br/>  for <font color="Olive" title="b3">f</font> being   <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> (<font color="Olive" title="b2">o</font>,<font color="Maroon" title="c2">r</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> (<font color="Olive" title="b2">o</font>,<font color="Maroon" title="c2">r</font>)</span>)</span>  st <font color="Olive" title="b3">f</font> <a href="abian.html#NR4" title="ABIAN:NR.4">has_no_fixpoint</a>  holds <br/><span class="p1">(<span class="default"><a href="brouwer.html#K5" title="BROUWER:func.5">BR-map</a> <font color="Olive" title="b3">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<font color="Olive" title="b2">o</font>,<font color="Maroon" title="c2">r</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <span class="p1">(<span class="default"><a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> (<font color="Olive" title="b2">o</font>,<font color="Maroon" title="c2">r</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">n</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">o</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c3">n</font></span>)</span><br/>  for <font color="Olive" title="b2">f</font> being   <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> (<font color="Olive" title="b1">o</font>,<font color="Maroon" title="c2">r</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> (<font color="Olive" title="b1">o</font>,<font color="Maroon" title="c2">r</font>)</span>)</span>  st <font color="Olive" title="b2">f</font> <a href="abian.html#NR4" title="ABIAN:NR.4">has_no_fixpoint</a>  holds <br/><span class="p1">(<span class="default"><a href="brouwer.html#K5" title="BROUWER:func.5">BR-map</a> <font color="Olive" title="b2">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<font color="Olive" title="b1">o</font>,<font color="Maroon" title="c2">r</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <span class="p1">(<span class="default"><a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> (<font color="Olive" title="b1">o</font>,<font color="Maroon" title="c2">r</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">o</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c3">n</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> (<font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c2">r</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> (<font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c2">r</font>)</span>)</span>  st <font color="Olive" title="b1">f</font> <a href="abian.html#NR4" title="ABIAN:NR.4">has_no_fixpoint</a>  holds <br/><span class="p1">(<span class="default"><a href="brouwer.html#K5" title="BROUWER:func.5">BR-map</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c2">r</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <span class="p1">(<span class="default"><a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> (<font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c2">r</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">f</font> be   <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> (<font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c2">r</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> (<font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c2">r</font>)</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">f</font> <a href="abian.html#NR4" title="ABIAN:NR.4">has_no_fixpoint</a>  implies <span class="p1">(<span class="default"><a href="brouwer.html#K5" title="BROUWER:func.5">BR-map</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c2">r</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <span class="p1">(<span class="default"><a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> (<font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c2">r</font>)</span>)</span> )</span><br/>







<span class="kw">assume </span><a NAME="E1:20"/><span class="lab"><font color="Green" title="E13">A1</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="abian.html#NR4" title="ABIAN:NR.4">has_no_fixpoint</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="brouwer.html#K5" title="BROUWER:func.5">BR-map</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c2">r</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <span class="p1">(<span class="default"><a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> (<font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c2">r</font>)</span>)</span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c6">D</font> =  <a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> (<font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c2">r</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c7">C</font> =  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c2">r</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c8">g</font> =  <a href="brouwer.html#K5" title="BROUWER:func.5">BR-map</a> <font color="Maroon" title="c5">f</font>;<br/>
<a NAME="E2:20"/>
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="brouwer.html#K5" title="BROUWER:func.5">BR-map</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> (<font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c2">r</font>)</span>)</span> &amp;  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> (<font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c2">r</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> (<font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c2">r</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="brouwer.html#T3" target="_self" title="BROUWER:th.3">Th3</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E3:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A2</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="brouwer.html#K5" title="BROUWER:func.5">BR-map</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c2">r</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c2">r</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T62" title="RELAT_1:th.62">RELAT_1:62</a>, <a class="ref" href="topreal9.html#T17" title="TOPREAL9:th.17">TOPREAL9:17</a></span>;<br/>
<a NAME="E4:20"/><span class="lab"><font color="Green" title="E15">A3</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> (<font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c2">r</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c2">r</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="toprealb.html#T9" title="TOPREALB:th.9">TOPREALB:9</a></span>;<br/>
<a NAME="E5:20"/><span class="lab"><font color="Green" title="E16">A4</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="brouwer.html#K5" title="BROUWER:func.5">BR-map</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c2">r</font>)</span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="brouwer.html#K5" title="BROUWER:func.5">BR-map</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c2">r</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <span class="p2">(<span class="default"><a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> (<font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c2">r</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="brouwer.html#K5" title="BROUWER:func.5">BR-map</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c2">r</font>)</span>)</span></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="brouwer.html#K5" title="BROUWER:func.5">BR-map</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c2">r</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <span class="p2">(<span class="default"><a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> (<font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c2">r</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:20_1"/><span class="lab"><font color="Green" title="E17">A5</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="brouwer.html#K5" title="BROUWER:func.5">BR-map</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c2">r</font>)</span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="brouwer.html#K5" title="BROUWER:func.5">BR-map</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c2">r</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <span class="p2">(<span class="default"><a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> (<font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c2">r</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c10">y</font> = <font color="Maroon" title="c9">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> (<font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c2">r</font>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_1"><span class="lab"><font color="Green" title="E17">A5</font></span></a></span>;<br/>
<a NAME="E3:20_1"/><span class="lab"><font color="Green" title="E18">A6</font></span>: 
 not <font color="Maroon" title="c9">x</font> <a href="abian.html#R1" title="ABIAN:pred.1">is_a_fixpoint_of</a> <font color="Maroon" title="c5">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="ref" href="abian.html#D5" title="ABIAN:def.5">ABIAN:def 5</a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="brouwer.html#K5" title="BROUWER:func.5">BR-map</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c2">r</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> = 
<span class="p1">(<span class="default"><a href="brouwer.html#K5" title="BROUWER:func.5">BR-map</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E1:20_1"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>
<br/>.= 
<font color="Maroon" title="c10">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E4:20"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="txt" href="#E1:20_1"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="txt" href="#E3:20_1"><span class="lab"><font color="Green" title="E18">A6</font></span></a>, <a class="ref" href="brouwer.html#T11" target="_self" title="BROUWER:th.11">Th11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <span class="p2">(<span class="default"><a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> (<font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c2">r</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E4:20"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="txt" href="#E1:20_1"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="ref" href="funct_1.html#T18" title="FUNCT_1:th.18">FUNCT_1:18</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E6:20"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <span class="p2">(<span class="default"><a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> (<font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c2">r</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> (<font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c2">r</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T45" title="RELAT_1:th.45">RELAT_1:45</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:20"/>
<span class="p1">(<span class="default"><a href="brouwer.html#K5" title="BROUWER:func.5">BR-map</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c2">r</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <span class="p1">(<span class="default"><a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> (<font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c2">r</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E4:20"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="txt" href="#E5:20"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="ref" href="funct_1.html#T2" title="FUNCT_1:th.2">FUNCT_1:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
